Complement in Distributive Lattice is Unique/Corollary
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Corollary to Complement in Distributive Lattice is Unique
Let $\struct {S, \vee, \wedge, \preceq}$ be a Boolean lattice.
Then every $a \in S$ has a unique complement $\neg a$.
Proof
By definition, a Boolean lattice is a complemented distributive lattice.
A complemented lattice is bounded by definition.
By Complement in Distributive Lattice is Unique, it follows that every $a \in S$ has at most one complement.
Since $\struct {S, \vee, \wedge, \preceq}$ is complemented, the result follows.
$\blacksquare$